0387986995Basic TheoryC

Problem 1-3-6. Show that for any fixed t > 0, there exists a real-valued function 0(x) such that

(1)0(x) is continuous on an interval 0 < C-x <- 6 for some small positive number 6 which depends on (,

(2)fix) is continuous on an interval 0 < - z < 6,

(3)the function

1 + (: - x)4(x)

y(x) -

F( - x)

satisfies the differential equation

ono<x<6.

Answer.

Step 1. Ifwesett=£-randy = , the given differential equation is changed

(1)t2u" - 2tu' + 2u + 9(tu' - u)u = 2t(tu' - u)u + tsu-3 - tut,

where f = dt

=w2,

_ - 2(1 + wl) + 3w2 - 2(w2 - (1 + wl))(1 + w1)

(2)

+ 2r1t(w2 - (1 +wl))(1 + w1)

If we set t = 0, wl = 0, and w2 = 0, the right-hand side of (2) vanishes. Now, (2) can be written in the form

td!W--' = W2, dt

t dw2 =2w1+w2-2w1(w2-wl)

0 < t < to. To do this, first note that, if IZ(t)I < K on 0 < t < to, we obtain

t

= Iclt+t (' - 1 )Ko+t (In (12)) LK

< Iclto + K0 + e-1 t0LK.

For a positive number co, fix another positive number K so that co + Ko < 2 and

then fix t0 > 0 so that to < 1, toL < 1, and toKIPI < 1. Here, IPI = max{IP.kl

1 < j, k < 2}, where P = (Pjkj? k=1. Define successive approximations by

to

It can be shown that the successive approximations converges uniformly for Icl < Co and 0 < t < to. This establishes the existence of a bounded solution of the integral equation (8). Thus, we can complete the construction of solution (Sol) of equation

(Eq).

1-4. Analytic differential equations

So far, it has been assumed that the independent variable t and the unknown quantity y are real. Since a complex number is a two-dimensional real vector, every result in the previous sections can be extended to the case when the entries of g' and of the function f (t, g) are complex. Furthermore, if the function f is analytic with respect to (t, yam), we can construct analytic solutions by means of successive approximations. Denote by C the set of all complex numbers. The set of all n-

column vectors with complex entries is denoted by C".

by using the successive approximations

Oo(z) = 2 ,

01(z) = 2 + 4z - 2z2,

9 + 9 - 63

Combining Theorem 1-3-5 and Theorem 1-4-1, we obtain the following theorem:

Theorem I-4-6. If all the entries of an n x n matrix A(t) and C"-valued function b(t) are analytic on an interval Z, then every solution of linear differential equation (1.3.1) is analytic on Z.

EXERCISES I

I-1. Solve the initial-value problem

Lipschitz condition

If(x,yl)-f(x,y2)I < IY1Y2I

1-3. Show that the initial-value problem

has one and only one solution on the interval Ix - 51 < 7.

1-4. Suppose that u(x) is continuous and satisfies the integral equation

x

u(x) = J sin(u(t))u(t)pdt

0

on the interval 0 < x < 1. Show that u(x) = 0 on this interval if p > 0. What would happen if p < 0?

u-p

Hint. In case p < 0, the problem is reduced to the initial-value problem sin(u) du = dx, u(O) = 0. Thus, the solution u(x) is not identically zero and has a branch point at x = 0.

1-5. Show that if real-valued continuous functions f (x), g(x), and h(x) satisfy the inequalities

1-6. Let a(t) be a real-valued function which is continuous on the interval I = It:

0 < t < 11. For every positive integer rn, set

for some positive number K. Also

fta(r)dr-Lmn(mj-(t-mja(ml)`

E,,,

1-7. Let f (t, y) be a real-valued function of two independent variables t and y which is continuous on a region:

D = {(t, y) : 91(t) 5 y <- 92(t), r1 < t < r2}.

Suppose that

(i) 91, g'1, 92, and g2 are continuous on rl < t < T2,

(ii) 91(t) < 92(t) on rl < t < T2,

(iii)

(iv) Tl < t0 < r2 and g1(to) < CO < 92(to)-

Show that there exists a function 0(t) such that

(1)4 and 0' are continuous on to < t < T2 and (t,4i(t)) E D on to < t

(2)Q'(t) = f (t, O(t)) on to < t < 72 and 0(to) = co.

Also, let rt(x) be an arbitrary continuous function of x on an interval 0 < x < xo, where xo is a positive number. Define successive approximations by

Show that lim ym(x) exists uniformly on the interval 0 < x < xo. m-+oo

Hint. For this problem, see [Na3, pp. 143-160; in particular Bemerkung 2].

1-9. Let B be a Banach space over the field R of real numbers. Also, let f (t, x) be a B-valued function of the variables (t, x) E R x B. Consider an initial-value problem

where (r, {) is a given point in R x B. Assume that f satisfies the following condi-

tions:

(i) f(t,x)iscontinuous onaregion R={(t,x)ERxB:It-rl <a, IIx-t:ll<b},

where a and b are positive numbers and II II is the norm of B,

(ii) f(t.x) also satisfies the Lipschitz condition IIf(t,xl) -f(t,x2)II 5 LIIX1 -X211 whenever (t, xl) E R and (t, x2) E R, where L is a positive constant.

Show that

(1) there exists a positive number Af such that Ilf(t,x)II < M for (t,x) E R,

b

Hint. See [Huk7].

I-10. Let B be a Banach space over the field R of real numbers and let C be a nonempty compact subset of B. Also, on a bounded interval Z, let F be an infinite and equicontinuous set of C-valued functions. Show that F contains an infinite sequence which is uniformly convergent on Z.

I-11. Let B be a Banach space over the field R of real numbers and let C be a nonempty compact and convex subset of B. Assume that a C-valued function f (t, x) is continuous on a region

R={(t,x)ERxB:It-rl <-a,llx-t1l <-b},

where r E R, t E B, a > 0, b > 0, and 11 11 is the norm of B. Show that

(1) there exists a positive number M such that IIf(t,z)115 M for (t,x) E R,

Hint. Use a method similar to that of Remark 1-2-7. See, also, [Huk7l.

1-12. Let f (t, x") be Re-valued and continuous for (t, x") E Rr+t. Assume also that

E is a nonempty closed set in R"+t. Show that in order that for every (r,)) E E, there exists a real number a(r, depending on (r, ) such that r < a(-r, {) and

that the intial-value problem di = f (t, xF), x(r) _ has a solution s = ¢(t, r,

satisfying the condition (t, ¢(t, r, ) E E on an interval r < t < o(r), it is necessary and sufficient that for every (r, f E E and every positive number e, there exists a point (a, S) E E such that

where (a, ) generaly depends on (r, {, e).

Hint. The necessity is easy to prove. For sufficiency, use an idea similar to that of

Remark 1-2-7. See, also, [Huk6].

1-13. Assume that f (x, yt, y2,... , y,,) is real-valued, continuous, and continuously

b, lyj I < r (j = 1,... , n)), where a and b are real numbers such that a < b, and r is

a positive number. Assume, also, that (1) c is a real number such that a < c < b,

l

(2) 6(x) is a real-valued solution of !° = f (x, y, Ly, ... , fn-Y) on an interval

lz - cl < p, where p is a positive number such that a < c - p < c + p < b, (3)

r (k = 0,1,... n - 1). and (4) f(x,0,... ,0) = 0 on a < x < b.

dx*

Show that if there eixsts a sequence {ch : h = 1, 2, ... } of real numbers such that

Um ch = c and that ¢(ch) = 0 (h = 1, 2, ... ), then O(x) = 0 identically on h_+00

a<x<b.

1-14. Let u(x, y) be a continuously differentiable function of two independent variables (x, y) in a domain D = {(x, y) : y > 0, a + Ay < x < b - Ay}, where A is a positive number and the two quantities a and b are real numbers such that a < b.

Assume that u(x, y) is continuous on the closure f) of D and satisfies the condition

< AI e I +BluF+C

on D, where B and C are positive numbers. Set M = max(Iu(x,0)(: a < r < b}.

Show that

u(x, y) 1 5 Me 11 + (eBV - 1)

Hint. The last inequality is the Haar inequality. (See [Haa1, [Na5, pp. 51-561, and

[Har2, pp. 140-141]).

1-15. Let H(u, t, x, q) be a function defined on an open set in R4 containing the point (u, t, x, q) = (0, 0, 0, 0) and satisfy a uniform Lipschitz condition with respect to (u, q). Let O(x) be a function of class C' satisfying 0(0) = 0 and 0=(0) = 0.

Show that the initial-value problem

ut + H(u, t, x, u=) = 0, u(x, 0) = Q(x) has at most one solution of class C1 in a neighborhood of x = 0.

Hint. This is Haar's uniqueness theorem of the given partial differential equation

(cf. the references in Exercise 1-14). Upon applying Exercise 1-14, with M = C = 0, to the difference ti(t, x) = ul (t, x) - U2 (t, x) of two solutions ul(t, x) and U2 (t, x), the proof follows immediately.